By Leonid I. Piterbarg, Alexander G. Ostrovskii (auth.)

ISBN-10: 1441947736

ISBN-13: 9781441947734

ISBN-10: 1475744587

ISBN-13: 9781475744583

This ebook originated from our curiosity in sea floor temperature variability. Our preliminary, even though solely pragmatic, target was once to derive enough mathemat ical instruments for dealing with convinced oceanographic difficulties. ultimately, even though, those concerns went a ways past oceanographic functions partially simply because one of many authors is a mathematician. We chanced on that many theoretical problems with turbulent shipping difficulties were time and again mentioned in fields of hy drodynamics, plasma and good topic physics, and arithmetic itself. There are few monographs curious about turbulent diffusion within the ocean (Csanady 1973, Okubo 1980, Monin and Ozmidov 1988). whereas identifying fabric for this e-book we concentrated, first, on theoretical concerns which may be worthy for realizing mix procedures within the ocean, and, sec ond, on our personal contribution to the matter. Mathematically all the concerns addressed during this ebook are focused round a unmarried linear equation: the stochastic advection-diffusion equation. there is not any try and derive common statistics for turbulent circulation. in its place, the focal point is on a statistical description of a passive scalar (tracer) below given pace statistics. As for functions, this publication addresses just one phenomenon: shipping of sea floor temperature anomalies. optimistically, notwithstanding, our major methods are appropriate to different subjects.

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**Example text**

7 e = "-e \7 2 e. 42) Btl where Ue r/) I I j3(c) (tl (t ,r ) = a(c) u a(c)' j3(c) , "-e j3(c)2 = a(c) "-. 43) Such a transformation of variables is called the rescaling or renormalization. It is clear that in this situation all the time scales are functions of c only and hence any scale separation completely determines the asymptotical behavior of the mean tracer. 39) can be obtained by choosing a(c) = c 2, 37 Scale Classification j3(c) = c. 44) is the following statement on homogenization. 27).

2 MAIN TIME SCALES IN THE TURBULENT DIFFUSION PROBLEM In this section as a basis for unifying different physical and mathematical approaches to turbulent diffusion we suggest 'scale classification'. For the sake of simplicity, we restrict our consideration to homogeneous velocity fields with only one time scale (correlation time) and only one space scale (correlation radius). Therefore in the simplest case there are four independent time scales: • the observation time (we consider a Cauchy problem), • the molecular diffusion time, • the correlation time, • the turnover time defined as the ratio of the correlation radius and the mean square velocity fluctuation.

29 Scale Classification If all these scales have the same order of magnitude it is hopeless to attempt any analytical solution of the problem. Explicit formulas for the effective diffusivity or explicit equations for the statistical characteristics of the tracer can be obtained only if one or more of these scales is much smaller or much bigger than the others. We shall show that many well known exactly solvable models can be formulated in terms of the orders of these time scales. Our focus will be on the so called 'short-correlation approximation', which means that the correlation time of velocity is much smaller than both the observation time and the molecular diffusivity time.

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